Integrand size = 15, antiderivative size = 74 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=-5 a b \sqrt {-a+b x}+\frac {5}{3} b (-a+b x)^{3/2}-\frac {(-a+b x)^{5/2}}{x}+5 a^{3/2} b \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {43, 52, 65, 211} \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=5 a^{3/2} b \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-\frac {(b x-a)^{5/2}}{x}+\frac {5}{3} b (b x-a)^{3/2}-5 a b \sqrt {b x-a} \]
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Rule 43
Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = -\frac {(-a+b x)^{5/2}}{x}+\frac {1}{2} (5 b) \int \frac {(-a+b x)^{3/2}}{x} \, dx \\ & = \frac {5}{3} b (-a+b x)^{3/2}-\frac {(-a+b x)^{5/2}}{x}-\frac {1}{2} (5 a b) \int \frac {\sqrt {-a+b x}}{x} \, dx \\ & = -5 a b \sqrt {-a+b x}+\frac {5}{3} b (-a+b x)^{3/2}-\frac {(-a+b x)^{5/2}}{x}+\frac {1}{2} \left (5 a^2 b\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx \\ & = -5 a b \sqrt {-a+b x}+\frac {5}{3} b (-a+b x)^{3/2}-\frac {(-a+b x)^{5/2}}{x}+\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right ) \\ & = -5 a b \sqrt {-a+b x}+\frac {5}{3} b (-a+b x)^{3/2}-\frac {(-a+b x)^{5/2}}{x}+5 a^{3/2} b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=-\frac {\sqrt {-a+b x} \left (3 a^2+14 a b x-2 b^2 x^2\right )}{3 x}+5 a^{3/2} b \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(\frac {5 a^{\frac {3}{2}} b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) x -\sqrt {b x -a}\, \left (-\frac {2}{3} b^{2} x^{2}+\frac {14}{3} a b x +a^{2}\right )}{x}\) | \(55\) |
derivativedivides | \(2 b \left (\frac {\left (b x -a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x -a}+a^{2} \left (-\frac {\sqrt {b x -a}}{2 b x}+\frac {5 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\right )\) | \(69\) |
default | \(2 b \left (\frac {\left (b x -a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x -a}+a^{2} \left (-\frac {\sqrt {b x -a}}{2 b x}+\frac {5 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )\right )\) | \(69\) |
risch | \(\frac {a^{2} \left (-b x +a \right )}{x \sqrt {b x -a}}+\frac {2 b \left (b x -a \right )^{\frac {3}{2}}}{3}-4 a b \sqrt {b x -a}+5 a^{\frac {3}{2}} b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )\) | \(69\) |
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Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.77 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=\left [\frac {15 \, \sqrt {-a} a b x \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, {\left (2 \, b^{2} x^{2} - 14 \, a b x - 3 \, a^{2}\right )} \sqrt {b x - a}}{6 \, x}, \frac {15 \, a^{\frac {3}{2}} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (2 \, b^{2} x^{2} - 14 \, a b x - 3 \, a^{2}\right )} \sqrt {b x - a}}{3 \, x}\right ] \]
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Result contains complex when optimal does not.
Time = 3.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.31 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=\begin {cases} - \frac {a^{\frac {5}{2}} \sqrt {-1 + \frac {b x}{a}}}{x} - \frac {14 a^{\frac {3}{2}} b \sqrt {-1 + \frac {b x}{a}}}{3} - \frac {5 i a^{\frac {3}{2}} b \log {\left (\frac {b x}{a} \right )}}{2} + 5 i a^{\frac {3}{2}} b \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - 5 a^{\frac {3}{2}} b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {2 \sqrt {a} b^{2} x \sqrt {-1 + \frac {b x}{a}}}{3} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {i a^{\frac {5}{2}} \sqrt {1 - \frac {b x}{a}}}{x} - \frac {14 i a^{\frac {3}{2}} b \sqrt {1 - \frac {b x}{a}}}{3} - \frac {5 i a^{\frac {3}{2}} b \log {\left (\frac {b x}{a} \right )}}{2} + 5 i a^{\frac {3}{2}} b \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )} + \frac {2 i \sqrt {a} b^{2} x \sqrt {1 - \frac {b x}{a}}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=5 \, a^{\frac {3}{2}} b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} b - 4 \, \sqrt {b x - a} a b - \frac {\sqrt {b x - a} a^{2}}{x} \]
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Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=\frac {15 \, a^{\frac {3}{2}} b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} - 12 \, \sqrt {b x - a} a b^{2} - \frac {3 \, \sqrt {b x - a} a^{2} b}{x}}{3 \, b} \]
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Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85 \[ \int \frac {(-a+b x)^{5/2}}{x^2} \, dx=\frac {2\,b\,{\left (b\,x-a\right )}^{3/2}}{3}-\frac {a^2\,\sqrt {b\,x-a}}{x}+5\,a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )-4\,a\,b\,\sqrt {b\,x-a} \]
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